Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

Problem 6.3.20a

Chapter 6, Problem 6.3.20a

Alcohol-Impaired Driving You wish to estimate, with 95% confidence, the population proportion of motor vehicle fatalities that were caused by alcohol-impaired driving. Your estimate must be accurate within 5% of the population proportion.
a. No preliminary estimate is available. Find the minimum sample size needed

Verified step by step guidance

Determine the formula for the minimum sample size required to estimate a population proportion with a given margin of error. The formula is:

n_{\min} = \frac{z^{2}}{E ²} × 0.25, where z is the z-score corresponding to the confidence level, E is the margin of error, and 0.25 is used when no preliminary estimate is available.

Identify the z-score for a 95% confidence level. For a 95% confidence level, the z-score is approximately 1.96. This value corresponds to the critical value of the standard normal distribution for a two-tailed test.

Substitute the margin of error (E) into the formula. The problem specifies that the estimate must be accurate within 5%, so

E = 0.05 .

Substitute the values of

z

and

E

into the formula:

n_{\min} = \frac{{1.96}^{2}}{0.05 ²} × 0.25 .

Simplify the expression to calculate the minimum sample size. Ensure that the result is rounded up to the nearest whole number, as sample size must be a whole number.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Population Proportion

The population proportion refers to the fraction of a population that exhibits a certain characteristic, in this case, the proportion of motor vehicle fatalities caused by alcohol-impaired driving. Understanding this concept is crucial for estimating how widespread an issue is within a larger group, which informs decisions and policies.

Sample Size Determination

Sample size determination is the process of calculating the number of observations or replicates needed to ensure that the sample accurately reflects the population. In this context, it involves using statistical formulas to find the minimum number of fatalities required to estimate the population proportion with a specified level of confidence and margin of error.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence, such as 95%. This concept is essential for understanding the reliability of the estimate, as it indicates how much uncertainty is associated with the sample data.

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Introduction to Confidence Intervals

Related Practice

Textbook Question

Constructing Confidence Intervals In Exercises 13–24, assume the sample is from a normally distributed population and construct the indicated confidence intervals for (b) the population standard deviation σ. Interpret the results.

Drug Concentration The times (in minutes) for the drug concentration to peak when the drug epinephrine is injected into 15 randomly selected patients are listed. Use a 90% level of confidence.

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Textbook Question

Drive-Thru Times The times (in seconds) spent by a random sample of 28 customers in the drive-thru of a fast-food restaurant have a sample standard deviation of 56.1. Use a 98% level of confidence.

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Textbook Question

Congress You wish to estimate, with 95% confidence, the population proportion of likely U.S. voters who think Congress is doing a good or excellent job. Your estimate must be accurate within 4% of the population proportion.

a. No preliminary estimate is available. Find the minimum sample size needed.

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Textbook Question

Soccer Balls A soccer ball manufacturer wants to estimate the mean circumference of soccer balls within 0.15 inch.

a. Determine the minimum sample size required to construct a 99% confidence interval for the population mean. Assume the population standard deviation is 0.5 inch

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Textbook Question

Constructing Confidence Intervals In Exercises 13–24, assume the sample is from a normally distributed population and construct the indicated confidence intervals for (a) the population variance σ^2. Interpret the results.

Volleyball The numbers of service aces scored by 15 teams randomly selected from the top 50 NCAA Division I Women’s Volleyball teams for the 2021 season have a sample standard deviation of 26.1. Use an 80% level of confidence. (Source: National Collegiate Athletic Association)

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Textbook Question

Ages of College Students An admissions director wants to estimate the mean age of all students enrolled at a college. The estimate must be within 1.5 years of the population mean. Assume the population of ages is normally distributed.

a. Determine the minimum sample size required to construct a 90% confidence interval for the population mean. Assume the population standard deviation is 1.6 years.

102

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